If the District budget says Congress authorized the employment of 34,797 people this year and the city has left more than 3,000 of those positions unfilled to save money, how many paychecks were issued in the mid-September pay period? The answer is 43,419.

The apparent discrepancy in those numbers can be explained and there is no scandal involved. Some workers paid directly out of federal grants show up on the payroll but not the budget; other checks go to workers known as WAE (when actually employed), who are paid for part time or irregular work. And some people are paid twice, legally.

But the payroll exemplifies the difficulty encountered by city officials, Congress, the press and the public in pinning down just how much money the financially squeezed city government is spending and for what purposes.

"I know the average citizen in the street just can't conceive of this," said Raymond Saulino, chief of the city's pay and retirement office. "He can't imagine that the number of people on the payroll isn't the same as the number of paychecks, but that's the way the system works."

The payroll system that Saulino inherited when he took the job in April is a hodgepodge that developed over years of governmental reorganizations, departmental and institutional mergers and sheer habit that took on a life of its own.

The workers are divided into three groups designated, for no known reason, 1, 2 and 6. Those in group six, mostly school teachers, are paid twice monthly, or 24 times a year. Those in groups one and two are paid biweekly, or 26 times a year, but not on the same days.

Employes of the University of the District of Columbia can be found in all three groups; workers in the Department of Environmental Services are in groups one and two; everybody who works for the public libraries is in group one. All the groups include, Saulino said, "a lot more than the full-time regular on-board career employes" who have traditionally been the only people counted in analyses of the city's work force.

The City Council and Congress, in their annual review of the District's budget, have focused on the full-time workers paid from appropriated funds. There are about 31,000 of them now, and their ranks are thinning as the city's financial crisis forces Mayor Marion Barry to trim the work force.

But another 6,000 are paid directly from federal grants. That is money given to the city by the federal government for housing, recreation, health, welfare and environmental programs administered by District agencies. Workers paid out of these funds are generally not subject to the budget review process, but they get their paychecks from the city like the workers who are.

Saulino said there are some people who receive more than one paycheck in each pay period.The biggest group is public school janitors, paid out of school funds, who also work part time for the recreation department and so get a second check, charged against a different account. Another smaller group consists of doctors at D.C. General Hospital who are more than one staff, such as residents who also work in the emergency room.

The WAEs, according to Saulino, represent a sizeable number of the monthly paychecks but small amounts of money, because many of them work only a day or two in a month. The Armory Board, he said, has workers who are called in only when an event is scheduled and paid only when they work. Members of city boards and commissions who are paid only for meetings they attend are also in this category. "They're not that significant in terms of total dollars," he said.

In the summer of 1979, the number of people paid on District checks rose to about 65,000 because the temporary summer youth workers were included. This year, those youths were hired and paid by a private contractor -- much to the relief of Saulino and his staff, who were spared from wrestling with the problems of a trouble-plagued program in which checks were delayed or missed.

The apparent discrepancy between the size of the payroll discussed in public forums and the number of checks actually issued is not surprising to city officials, who take the grant funds workers and the WAEs for granted. "That's about the right number," said City Auditor Matthew S. Watson, the city council's fiscal watchdog. "You could question whether the payroll ought to be that big, but 43,000 is about what you would expect."

The payroll statistics do bear out Barry's claims to have reduced overall city employment; in the comparable pay period a year ago, 45,912 checks were issued. But the computer printouts summarizing the payroll do not indicate how many of the checks being issued are going to full-time workers, how many to part-timers and how many to WAEs. Saulino said he hoped to begin getting that information in November.